Is there a continuous version of the Borel-Cantelli lemma? Given a sequence of events $A_n$ for  $n\in \mathbb N$, the first Borel Cantelli lemma states that, if the sum over all probabilities $\sum_{i=1}^n P(A_n)$ is finite, then the probability of the limit supremum of the $A_n$ is zero. Or more intuitively, if the sum $\sum_{i=1}^n P(A_n)$ is finite, then the probability for $\{ A_n$ happens infinitely often } is zero.
Now my question is: is there a continuous version of this? 
i.e. is there any statement like: given $A_t$ with $t>0$, if $ \int_0^{\infty} P(A_t) dt<\infty$ then $P\{ A_t$ happens infinitely often }$=0$? 
 A: Let $T$ be the collection of all the $t$ that $A_t$ happens. That is, for any $\omega$, an element in the probability space which your are working with, define 
$$
T(\omega) := \{t\geq 0: \mathbf 1_{A_t} (\omega)=1\}.
$$
Denote $l(T)$ as the Lebesgue measure of this random set $T$.
Then, by Fubini's theorem, 
$$
\int_0^\infty \mathbf P[A_t]dt = \mathbf E[l(T)].
$$
Therefore, if the left side is finite, we surely have that
$$
l(T)<\infty
$$
happens as surely.
A: This discussion is independent of my first answer, which I still think is the right form of the continuous version of the Borel-Cantelli lemma.
However, it seems that people are interested in the following question:
"What is the right condition making a family of random events $(A_t)_{t\geq 0}$ as surely stop happening while $t$ is large enough?"
Unfortunately, as far as I know, there is no universally applicable theory that addresses this. 
Even if each event $A_t$ has zero probability, it is still not enough to allow us to make the conclusion that $(A_t)_{t\geq 0}$ will stop happening eventually. 
For example, let us consider the one demensional Brownian motion $B_t$, and define that $A_t:=\{B_t=0\}$.
It's easy to see that, for each $t>0​$, event $A_t​$ has probability 0. Which seems to imply that $A_t​$ should never happen.
However, as a matter of fact, a classical property of one demensional Brownian motion is that
$$
\limsup_{t\to\infty} B_t =\infty, \quad \liminf_{t\to\infty} B_t=-\infty.
$$
This says that $A_t$ will not stop happenning. 
So my opinion is that those type of problems should be discussed case by case.  
And often, those discussions will be related to the regularity property of the path of the indicated process:
$$
X_t:= \mathbf 1_{A_t}, \quad t\geq 0.
$$
In the Brownian motion example, the indicated process $X_t$ is very irregular. And as a consequence, the probability of events $A_t$ gives no information in answering the question.
In the case, that $X_t$ is a continuous process. We see that $A_t$ happens for all $t$ if and only if $A_0$ happens. So the desired property is solely determined by the probability of $A_0$.
In the case, that $X_t$ is a cadlag process. If we know that the probability of $A_t$ decay to 0 fast enough, then we should have that $(A_t)_{t\geq 0}$ will stop happening eventually. (Warning: I am not very sure about this last assertion.)
A: $\newcommand{\R}{\mathbb{R}}$
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$\newcommand{\P}{\operatorname{\mathbb{P}}}$
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This question can make sense if we frame it in the form of random variables. A stochastic process $\wrap{ X_t }_{t\in\R_+}$ is said to converge $X_t\overset{as}{\longrightarrow}X$ if $\P\Omega_0=1$ for:
$$\Omega_0\triangleq\set{\omega\in\Omega}{\lim_{t\rightarrow\infty}X_t(\omega)=X(\omega)}$$
We ask about Borel-Cantelli in the continuous case. If we have for all $\epsilon>0$:
$$
\int_{\R_+}dt\, \P\wrap{\abs{X_t-X}>\epsilon}<\infty
$$
Then do we have $X_t\rightarrow X$?
First, let's note the implications of this formulation for events. If $X_t=1_{A_t}$ for some events $A_t$ and $X=0$ then the above formulation is equivalent for any $\epsilon$ to OP's condition $\int_{\R_+}dt\,\P A_t<\infty$. And the conclusion $1_{A_t}\rightarrow 0$ indeed says that with probability 1 for sufficiently large $T$, $\bigcup_{t>T}A_t$ doesn't happen.
Unfortunately, this certainly does not hold. First, consider $X=0$ and:
$$
X_t=\begin{cases}
1&t\in\mathbb{Q}\\
0&\text{o/w}
\end{cases}
$$
Certainly for any $\epsilon$ the above integral is 0, though $X_t\not\rightarrow X$. One could require continuity in $t$. Still not enough. Let $g$ be some smooth unit window function with compact support (e.g., the Hann window). Then set $f_a(x)=g(ax)$, which has area $1/a$, and define:
$$
f(t)=\sum_{n=1}^\infty f_{n^2}(t-n)
$$
Then setting $X_t=f(t)$ and $X=0$ we have the same issue as above.
The issue is that the finite integral condition obscures information about Lebesgue-negligible sets in $\R$, but these sets can misbehave with the topological condition $\lim_t X_t=X$.
A: This is just a clarification of Zhenyao Sun's answer (measurability and why this generalization is natural) 

Let $A \subset [0,\infty)\times \Omega$ be measurable and $A_t = \{\omega\in \Omega: (t, \omega) \in A\} $.  If 
  $$
\int_0^\infty \mathrm P(A_t) dt<\infty, 
$$
  then $$\lambda(\{t\ge 0: A_t \text{ happens} \}) <\infty$$ almost surely. 

Proof  By Fubini's theorem, 
$$
\int_0^\infty \mathrm P(A_t) dt = \mathrm E[\lambda(\{t\ge 0: A_t \text{ happens} \})] <\infty, 
$$
whence the statement follows. 

Why is it natural? If $[0,\infty)$ is replaced by $\mathbb N$ and $\lambda$, by the counting measure, then it transforms to the usual Borel-Cantelli lemma. 
